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What's math got to do with it?
Numeracy & Social Studies Education
OCSS 2009
Alicia R. Crowe
Kent State University
March 26, 2009
acrowe@kent.edu
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Perhaps H. G. Wells was right
when he said statistical
thinking will one day be as
necessary for efficient
citizenship as the ability to
read and write!
•
S. S. Wilks (1951)
“The time may not be very remote
when it will be understood that for
complete initiation as an efficient
citizen of one of the new great
complex world-wide states that are
now developing, it is as necessary to
be able to compute, to think in
averages and maxima and minima, as
it is now to be able to read and write”
H. G. Wells (1904)
•
Just as understanding what
words mean and how they are
being used in a specific context,
the understanding of numeric
information and how it is being
used is vitally important when
making decisions on issues that
impact the quality of life of
ourselves and others in our
society
Examples
understand economic data and make
decisions about what to do (i.e. buy a house,
sell your house, take out a loan) and what to
ask (i.e. Is this interest rate appropriate or is
this predatory lending? Does the interest
rate hike impact my daily life?);
understand scientific or medical information
and make decisions about whether there is a
risk to taking a drug;
Examples
understand polling data and ask critical
questions about the way the poll was
administered, who was sampled, and how
the sampling occurred
understand data presented by politicians,
government offices, or corporations as they
make a case for a specific policy
understanding the risks of markets
•
Basically, certain aspects of
mathematics and mathematical ways
of thinking are fundamental to
understanding social studies concepts
and becoming active members of
society.
Four Instances
1) the ability to understand raw numeric data
or numeric information in context;
2) the ability to understand percentages in
context;
3) an understanding of the meaning of
average; and
4) the ability to interpret and question graphs
and charts.
Numeric Data
A million, a billion, a
trillion - so1,000,000
what?
= 1 Million
•
•
•
1,000,000,000 = 1 Billion
1,000,000,000,000 = 1 Trillion
•
How many million in a billion?
•
What happens to our ability to compare when we
simplify 1,000,000,000,000 into 1 Trillion and
475,000,000 into 475 Million?
•
What
A million, a billion, a
trillion - so what?
•
As part of your decision-making process as a
citizen it is important to know the magnitude of
what you are supporting.
Where misunderstanding
can happen
In 2004, a newspaper article from the China Daily shared that
136,340 deaths occurred in industrial accidents in the
previous year (Xing, 2004). In the United States, , the Bureau
of Labor Statistics reported 5,559 “fatal occupational injuries”
in 2003 (BLS, 2004).
How many deaths are there in each country for the number of
workers there are (a ratio of number of deaths to the whole
population)?
How does each country count deaths?
What is the difference between deaths from “industrial
accidents” and “fatal occupational injuries?” Does it have to be
at the work-site to count in the United States? How does each
country count deaths from disease caused by the workplace?
Population
World Population
Ohio Population
U.S. Population (#3)
6,760,000,000
11,459,011
306,000,000
China (#1)
1,329,740,000
India (#2)
1,160,139,960
How many U.S. populations would it take to
equal that of China?
Why does this matter?
and What to do
•
As a citizen and a decision maker, a student should be
developing an understanding of the magnitude of numbers,
the value of the context of raw numbers, and the
importance of the relationships among numbers being
discussed as they learn to make decisions about what to
support.
Highlight the differences among numbers in things that are read
or numbers you present
Asking questions that help students to see the raw number in a
larger context, and
Point out relationships between raw numbers and percentages
when you use them or they are in the readings.
Percentages
Jeff and Jessie come home from school. Both
read an article discussing population growth
that day. At supper they are discussing it.
Jessie claims that population growth isn't
really a problem because population growth
has steadily decreased over the last 40 years
going from about 2% per year in 1960 to
1.5% in 1990 to an estimated 1% in 2015.
Jeff argues no, that just because the
percentage decreases does not mean that
there isn't still a significant increase in
population, and thus an area for concern –
the population is still growing too fast.
How do two people both look at the
same information and have two
different readings?
•
Percentages can be deceiving out of context
just like raw numbers can be deceiving out of
context.
What does this mean for understanding social
studies concepts?
The reference point matters
When the government announces a rise in
the unemployment rate citizens should ask
themselves - “What exactly does this
mean?” (January the rate was 7.6% and in
February it was 8.1%)
What does the rise signify?
How many people is this really?
What does this look like historically?
Check out the Bureau of Labor Statistics for this
info! http://www.bls.gov/
What can we do?
To start - embed questions about the ways in
which percentages and raw numbers can be
deceiving into everyday classroom
conversations about readings from the text,
newspaper articles, government documents, or
other readings you have included for your class.
What can we do?
In a U.S. Government class have the students bring in
newspaper articles that reflect a connection to the current
topic of study. In addition to the questions normally asked,
add questions like: Where are places in your newspaper
article that numbers are used? What does this data tell
you? Is there more data that would make this easier to
understand?
In US History or US Government course that includes a
focus on the creation of the Constitution and the bicameral
legislative branch you could use the issue of
representation (large state population vs. small state
population) as an example of percentages and a way to
bring mathematics in as you study the American
Revolution.
When numbers and percentages
The population
City A was 40,000 in
comeintogether
1990 and 50,000 in 2000. The population
in its county rival, City B was 60,000 in
1990 and 70,000 in 2000.
City A claims in their literature to have
grown more than any other area in the
county.
When are they right?
City B argues that this is untrue, that they
have grown equally.
A Quick Activity
Explore current news to find the
need for mathematical
understanding.
Describe where the
misunderstanding could occur
and possible results of the
misunderstanding.
Meaning of Average
The Meaning of Average
There are at least three ways to describe the
“center” of a set of data
the mean (arithmetic average)
the median (the center of the data if lined
up in order)
the mode (the most common figure)
What does it matter?
Historical data like census record
mean income vs median income
Current Topics
Campaign finances
How much did the “average” contributor give?
If I am running for a small local office and 99
people give $100 and 1 person gives $10,000
- the mean will be $199. How might this
mislead perceptions if the mean is used? The
median or the mode better represents the
“average” contributor.
An Example
Reports from local, state and federal government agencies
in the United States often use both median and mean as
they discuss average and some reports use only the term
average and do not clarify.
This can be confusing if as a citizen we automatically think
of all of these terms meaning our everyday definition of
average.
For example, if a community or county included mostly
households with incomes from $35,000 to $65,000 but
happened to have two or three households with incomes
in the millions, the mean would be so high that it could
mislead someone while the median would best represent
the income of the middle of the area.
•
Helping
Students
The U.S. Census Bureau collects and reports data on both
mean and median household income
•
For example, you could use their table
http://www.census.gov/hhes/www/income/histinc/h06ar.ht
ml as a “bell-work” item to begin a lesson that related to
understanding changes in the United States in the last 30
years.
•
With the chart projected on the screen in the room, where
both the median and mean are next to one another,
students could be asked to examine the chart and write
some questions they have about the data, then, the class
could discuss their questions and begin to hypothesize
why there would be a nearly $20,000 difference between
the median and mean income for 2006. and learning.
Helping Students
When the term “average” comes up, ask
students about what is meant in the context and
how they know.
When data is presented, have students figure
out mean and median to see which might might
represent a set of data
Interpret and Question
Graphs
Graphs do double duty
They reaches all learners in a different way
They helps students see change over time,
number, and proportion (for example)
They help students develop skills to be used in
everyday life as a citizen and
They help develop a richer understanding of
concepts, events, issues, etc.
Basic Types of Graphs
Bar Graph
Comparison among categories
Pie Chart
Comparison using percentages
Line Graph
To show change over time
Basic Elements to
Consider
Labels
Purpose
Scale and proportion
An example
It is often hard for an adolescent who lives in the United States to
visualize the impact of wars on the European or Asian continents
without the help of graphs and charts.
In a unit on World War II, pie charts could be show the devastating loss of life
both military and civilian.
Teacher or students could create pie charts showing the percentage of
population (military and civilian) who were not physically harmed, who were
killed, and who were wounded from Poland, Russia, and China (as three
examples) during the war to illustrate the impact of the war on these nations.
Or, this could be an opportunity to help build connections across time by
illustrating how smaller nations that are sometimes overlooked by our
textbooks were also impacted significantly by the war, Yugoslavia for example,
areas that come back in to the conversation in later years under study. With a
pie chart students would be able to compare these countries with very different
total populations while also seeing the magnitude of the devastation in each
country based on that country’s size.
Another example
If it is early in the year and your students are still not as comfortable with
questioning data then you might have them create and alter some graphs to begin
to see the differences. The National Council for Teachers of Mathematics
(http://illuminations.nctm.org/ActivityDetail.aspx?id=63) provides a useful tool that
helps to quickly show how manipulating the scale of a graph can alter our initial
perceptions of the data set. If you want to highlight how much the United States has
changed since the days of the creation of the Constitution then you might ask your
students to create a line graph of the United States population in ten-year
increments from 1790 to 2000.
To also teach about critically looking at data, you could ask some to use a
scale where 100,000 people were represented by one increment and have
other groups create the graph where 500,000 people were represented by the
same increment. Then discuss see how the use of scale can alter perceptions.
Starting simple
Problematize the textbook -- Start by modeling through questions
such as:
What does this chart really tell us?
Does this scale distort the data?
Is more information needed to make a decision?
Then, as the year progresses, help them to develop questions of
their own as they encounter data on a daily basis.
Problematize the news!
How might altering the scale on this chart change the way
that readers of this newspaper react to the issue?
How can we help students develop
mathematical literacy through social
studies?
Help them develop a certain level of statistical
understanding in the context of what they study
with us.
Help them learn to ask the right questions
Help them develop the habit of mind that
pushes them to consistently look beyond the
surface
Use mathematics in real world contexts that
support the learning of social studies content
For example
As a beginning activity early in the year, you could provide your
students with a set of questions to help them think about and
discuss a current or historical news piece as part of a larger
conversation about the content for the day.
What do the numbers in this piece mean? How could they be
represented differently? Where are the places that
misunderstanding can take place as a result of the author’s
use of numbers, percentages, or this chart or graph?
These can be expanded upon in discussions.
As the year progresses, add other questions, then begin to take
away the questions you ask and help them to develop their own
critical questions.
For example
Provide a policy document from a current or past
administration
Ask the students to develop a set of
questions that they wonder about based on
the document.
Then, add key questions you want about the
numeric data if they do not develop them
themselves.
Developing habits of
mind
Using the current news pieces you found as a
starting point, develop a set of basic questions
that you can pose in class that would help your
students develop their abilities to understand
numeric data.
What is the context in which the data is
presented?
Who is reporting the data?
What is the scale?
Connect to your
classroom
Come up with at least two ways
to explore the mathematical
aspects of social studies
understanding in your
classroom.
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Helpful Resources
Dewdney, A.K. (1993). 200% of nothing: An eye opening tour through the twists and turns
of math abuse and innumeracy. Hoboken, NJ: John Wiley & Sons, Inc.
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Gutstein, E., & Peterson, B. (eds.) (2005). Rethinking mathematics: Teaching social
justice by the numbers. Milwaukee, IL: Rethinking Schools.
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Huff, D. (1954). How to lie with statistics. New York: W.W. Norton & Company, Inc.
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Paulos, J.A. (1988). Innumeracy: Mathematical illiteracy and its consequences. New
York : Hill and Wang.
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Paulos, J. A. (1991). Beyond Numeracy: Ruminations of numbers man. New York:
Knopf.
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Paulos, J.A. (1996). A mathematician reads the newspaper. New York: First Anchor
Books.
Helpful Resources
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Data and Statistics - General Reference Resources: USA.gov:
http://www.usa.gov/Topics/Reference_Shelf/Data.shtml
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Statistics Canada: Canada’s National Statistical Agency:
http://www.statcan.ca/english/edu/power/ch9/using/using.htm
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Create-a-Graph: http://nces.ed.gov/nceskids/createagraph/default.aspx
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A review of bar graphs: http://cstl.syr.edu/FIPSE/TabBar/RevBar/REVBAR.HTM &
http://cstl.syr.edu/FIPSE/TabBar/BuildBar/BUILDBAR.HTM
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A review of of Pie Charts:
http://cstl.syr.edu/FIPSE/TabBar/ReadCirc/REVCIRCL.HTM &
http://cstl.syr.edu/FIPSE/TabBar/BldCirc/BUILDCIR.HTM